326 Mr. 0. Heaviside on the Electromagnetic Effects due 



The limits are such as include all space outside the sphere 

 r-=a. The coefficient ^ replaces •^. 



4. Next, as regards the mutual magnetic energy M of the 

 moving charge and any external magnetic field. This is the 

 space-summation ^|U.oHoH/47r, if Hq is the external field ; and, 

 by a well-known transformation, it is equivalent to S AqF, if Aq 

 is any vector whose curl is /aqHo, whilst V is the current-density 

 of the moving system. Further, if we choose Aq to have no 

 divergence, the polar part of F will contribute nothing to the 

 summation, so that we are reduced to the volume-integral of 

 the scalar product of the divergenceless Aq of the one system 

 and the density of the convection-current in the other. Or, 

 in the present case, with a single moving charge at a point, 

 we have simply the scalar product AoU^- to represent the 

 mutual magnetic energy ; or 



M=Aoii^, (3) 



which is double J. J. Thomson's result. 



5. When, therefore, we derive from (3) the mechanical 

 force on the moving charge due to the external magnetic 

 field, we obtain simply MaxwelPs "electromagnetic force" on 

 a current-element, the vector product of the moment of the 

 current and the induction of the external field ; or, if P is this 

 mechanical force, 



F=^o^ViiHo, (4) 



which is also double J. J. Thomson''s result. Notice that in 

 the application of the " electromagnetic force " formula, it is 

 the moment of the convection-current that occurs. This is 

 not the same as the moment of the true current, which 

 varies according to circumstances ; for instance, in the case 

 of a small dielectric sphere uniformly electrified throughout 

 its volume, the moment of the true current would be only 

 § of that of the convection-current. 



The application of Lagrange's equation of motion to (3) 

 also gives the force on q due to the electric field so far as it 

 can depend on M ; that is, a force 



dko 

 ~^-di' 



where the time-variation due to all causes must be reckoned, 

 except that due to the motion of q itself, which is allowed for 

 in (4). And besides this, there may be electric force not 

 derivable from Aq, viz. 



where 'SP'o is the scalar potential companion to Aq. 



